z^2+14z+27=0

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Solution for z^2+14z+27=0 equation: 


Simplifying
z2 + 14z + 27 = 0

Reorder the terms:
27 + 14z + z2 = 0

Solving
27 + 14z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-27' to each side of the equation.
27 + 14z + -27 + z2 = 0 + -27

Reorder the terms:
27 + -27 + 14z + z2 = 0 + -27

Combine like terms: 27 + -27 = 0
0 + 14z + z2 = 0 + -27
14z + z2 = 0 + -27

Combine like terms: 0 + -27 = -27
14z + z2 = -27

The z term is 14z.  Take half its coefficient (7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
14z + 49 + z2 = -27 + 49

Reorder the terms:
49 + 14z + z2 = -27 + 49

Combine like terms: -27 + 49 = 22
49 + 14z + z2 = 22

Factor a perfect square on the left side:
(z + 7)(z + 7) = 22

Calculate the square root of the right side: 4.69041576

Break this problem into two subproblems by setting 
(z + 7) equal to 4.69041576 and -4.69041576.


Subproblem 1
z + 7 = 4.69041576

Simplifying
z + 7 = 4.69041576

Reorder the terms:
7 + z = 4.69041576

Solving
7 + z = 4.69041576

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '-7' to each side of the equation.
7 + -7 + z = 4.69041576 + -7

Combine like terms: 7 + -7 = 0
0 + z = 4.69041576 + -7
z = 4.69041576 + -7

Combine like terms: 4.69041576 + -7 = -2.30958424
z = -2.30958424

Simplifying
z = -2.30958424


Subproblem 2
z + 7 = -4.69041576

Simplifying
z + 7 = -4.69041576

Reorder the terms:
7 + z = -4.69041576

Solving
7 + z = -4.69041576

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '-7' to each side of the equation.
7 + -7 + z = -4.69041576 + -7

Combine like terms: 7 + -7 = 0
0 + z = -4.69041576 + -7
z = -4.69041576 + -7

Combine like terms: -4.69041576 + -7 = -11.69041576
z = -11.69041576

Simplifying
z = -11.69041576


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-2.30958424, -11.69041576}

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